Diagonally Implicit RK Methods

Abstract

We continue to quote from this nice paper: “To integrate a system of n differential equations, an implicit method with a full s × s matrix requires the solution of ns simultaneous implicit (in general nonlinear) equations in each time step (...) One way to circumvent this difficulty is to use a lower triangular matrix (aij)(i.e., a matrix with aij
= 0 for i < j); the equations may then be solved in s successive stages with only an n -dimensional system to be solved at each stage”. In accordance with many authors, and in disaccordance with others (see above), we call such a method diagonally implicit (DIRK).

... they called their methods “diagonally implicit”, a term which is reserved here for the special case where all diagonal entries are equal ...